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Asai Gamma Factors and Distinction in families

Sabyasachi Dhar, Hariom Sharma

Abstract

Let $F$ be a finite extension of $\mathbb{Q}_p$ and let $E$ be a quadratic extension of $F$. A representation $(π,V)$ of ${\rm GL}_n(E)$ is said to be ${\rm GL}_n(F)$-distinguished if there exists a non-zero linear functional $φ$ on $V$ such that $φ(π(h)v) = φ(v)$ for all $h \in {\rm GL}_n(F)$ and $v \in V$. In this article, we study the notion of ${\rm GL}_n(F)$-distinguished representations for $R[{\rm GL}_n(E)]$ modules of Whittaker type, where $R$ is a Noetherian algebra over the ring of Witt vectors of $\overline{\mathbb{F}}_\ell$ with $\ell \ne p$. We first derive a functional equation, which gives the existence of the Asai $γ$-factors associated with $R[{\rm GL}_n(E)]$ modules of Whittaker type. We then provide a necessary condition for cuspidal $R[{\rm GL}_n(E)]$ modules of Whittaker type to be Whittaker ${\rm GL}_n(F)$-distinguished, expressed in terms of their Asai $γ$-factors.

Asai Gamma Factors and Distinction in families

Abstract

Let be a finite extension of and let be a quadratic extension of . A representation of is said to be -distinguished if there exists a non-zero linear functional on such that for all and . In this article, we study the notion of -distinguished representations for modules of Whittaker type, where is a Noetherian algebra over the ring of Witt vectors of with . We first derive a functional equation, which gives the existence of the Asai -factors associated with modules of Whittaker type. We then provide a necessary condition for cuspidal modules of Whittaker type to be Whittaker -distinguished, expressed in terms of their Asai -factors.
Paper Structure (8 sections, 19 theorems, 74 equations)

This paper contains 8 sections, 19 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Whittaker module
  4. Derivatives
  5. Asai zeta integrals and formal series
  6. Rationality of Asai Formal Series
  7. Functional equation
  8. Distinguished representations in families

Key Result

Theorem 1.0.1

Let $E$ be a quadratic extension of the $p$-adic field $F$. Let $R$ be a Noetherian $\Lambda$-algebra, and let $S$ be the multiplicative subset of $R[X,X^{-1}]$ consisting of Laurent polynomials whose first and last coefficients are units in $R$. Let $V$ be an $R[{\rm GL}_n(E)]$ module of Whittaker for all $W\in\mathbb{W}(V,\psi_E)$ and $\varphi\in C_c^\infty(F^n,R)$. Here, $\omega_V$ is the cent

Theorems & Definitions (37)

  • Theorem 1.0.1
  • Theorem 1.0.2
  • Definition 2.1.1
  • Lemma 3.0.1
  • proof
  • Lemma 3.0.2
  • proof
  • Lemma 4.0.1
  • proof
  • Corollary 4.0.2
  • ...and 27 more